A Class Of 3-D Chaotic System And The ?-Synchronization Scheme For Fractional-Order Systems

Chaotic phenomena are found in nonlinear systems commonly.Due to the chaotic system has special properties such as highly sensitive to initial conditions and has many unstable periodic orbits,nonlinear chaotic characteristics have been widely used in engineering and technology fields such as chaotic communication,secure communication,image encryption,power system protection,and DC motor control,etc.Therefore,more and more attention has been paid to the study of chaotic systems and their control.This thesis presents a three-dimensional chaotic system based on three-dimensional Lü chaotic system.Some basic dynamic behaviors of this chaotic system are analyzed,such as dissipativity,Lyapunov exponent spectrum,Poincaré section,and phase diagram,etc.It can be found that the three-dimensional chaotic system has two isolated chaotic attractors(herein referred to as "positive attractors" and "negative attractors"),which depend on the initial conditions and the distances between the unstable equilibrium points.And the necessary conditions for obtaining a "positive attractor" or "negative attractor" are given in this thesis.On the other hand,with the in-depth study of the fractional calculus theory,it is found that when the order of the chaotic system is a fraction,chaotic phenomenon still occurs.The three-dimensional chaotic system proposed in this thesis is extended to fractional order,and the dynamic behaviors of its fractional-order chaotic system are analyzed.The maximum Lyapunov exponents and chaotic attractors of the fractional-order chaotic system are studied.It is found that there are two isolated chaotic attractors in the fractional-order chaotic system,as the integer-order chaotic system.The same chaotic attractors rely on the initial points and the distances between the unstable equilibria.And through the simulation,the necessary conditions for generating "positive attractor" or "negative attractor" are the same as those for generating "positive attractor" or "negative attractor" by the integer-order system.In addition,for a class of fractional-order chaotic systems,a ?-synchronization scheme is proposed,its corresponding synchronization control strategy is given,and a strict mathematical proof of the scheme is put forward.This synchronization scheme is applied to the fractional-order chaotic system proposed and its two classic fractional-order chaotic systems in this thesis,and the numerical simulation is performed by using MATLAB.The simulation results are in good agreement with the theoretical analysis results,and the feasibility and effectiveness of the proposed ?-synchronization program are verified.